[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522874-105":59,"doc-detail-522874-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","relative-extrema-worksheets-4-relative-extrema","Relative Extrema Worksheets - 04 - Relative Extrema","","Worksheets for Infinite Calculus problems on finding relative minima and relative maxima for multiple functions. It includes polynomial exercises such as cubic and quartic expressions, plus a rational-style quartic form using shifted powers. Each problem asks for all relative extrema points, and solutions are provided as coordinate pairs (x, y), including cases with no relative maxima or no relative minima. The sheet also contains critical thinking questions requiring example functions satisfying f''(0)=0 with specified extrema behavior at x=0.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/relative-extrema-worksheets-4-relative-extrema/522874/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/relative-extrema-worksheets-4-relative-extrema/522874.png","ImageObject",300,407,{"name":92,"@type":93},"Sarah ","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-10","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How are relative minima and maxima found on these worksheets?","Question",{"text":112,"@type":113},"For each function, the worksheet asks for all points of relative minima and maxima and provides the corresponding coordinate pairs as answers.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What kinds of functions appear in the worksheet exercises?",{"text":117,"@type":113},"The problems focus on polynomial functions, including cubic and quartic forms, sometimes with shifts like (x − 4).",{"name":119,"@type":110,"acceptedAnswer":120},"What do the critical thinking questions require?",{"text":121,"@type":113},"They ask for example functions f(x) such that f''(0)=0, with either no relative minimum/maximum at x=0 or a relative maximum at x=0, and the worksheet lists example answers such as f(x)=0 and f(x)=−x4.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},522874,1791694191,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":52,"preview_page_count":4},962085320529,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Kuta Software-Infinite Calculus Name___________________________________  \nAssignment Date________________ Period____  \nFor each problem, find all points of relative minima and maxima.  \n1) y = x 3 − 5x2 + 7x − 5  \nFor each problem, find all points of relative minima and maxima. You may use the provided graph to sketch the function.  \n2) y = x 3 − 6x2 + 9x + 1  \n\n| | |  |  |  |  |  |  | y |  |  |  |  |  |  | |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  |  |  |  |  | 8 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 6 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 4 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 2 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n| −8 | − | 6 | − | 4 | − | 2 |  |  | 2 |  | 4 |  | 6 |  |  | 8 x |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −2 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −4 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −6 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  | −8\u003Cbr>|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n\nFor each problem, find all points of relative minima and maxima.  \n3) y = −x3 − 3x2 − 1 4) y = x 4 − 2x2 + 3  \n5) y = x 4 − x 2  2   \n6) y = − 2 x − 4  \n2 5 2  \n7) y = (2x − 8) 3  8) y = − ~~ ~~15(x − 4) 3 − 2(x − 4) 3  \nCritical thinking questions:  \n9) Give an example function f (x) where f '' (0) = 0 and there is no relative minimum or maximum at x = 0.  \n10) Give an example function f (x) where f '' (0) = 0 and there is a relative maximum at x = 0.  \nKuta Software-Infinite Calculus Name___________________________________  \nAssignment Date________________ Period____  \nFor each problem, find all points of relative minima and maxima.  \n1) y = x 3 − 5x2 + 7x − 5  \nRelative minimum: (~~ ~~73~~ ~~ , − ~~ ~~8627)  \nRelative maximum: ( 1, −2)  \nFor each problem, find all points of relative minima and maxima. You may use the provided graph to sketch the function.  \n2) y = x 3 − 6x2 + 9x + 1  \n\n| 8\u003Cbr>6 | y | |\n| --- | --- | --- |\n| 2 \u003Cbr>\u003Cbr>\u003Cbr>|  |  |\n\nRelative minimum: (3, 1)  \nRelative maximum: ( 1, 5)  \nFor each problem, find all points of relative minima and maxima.  \n3) y = −x3 − 3x2 − 1  \nRelative minimum: (−2, −5)  \nRelative maximum: (0, −1)  \n5) y = x 4 − x 2  \nRelative minima: (− ~~ ~~2~~ ~~, − ~~ ~~14),(~~ ~~2~~ ~~, − ~~ ~~14)  \nRelative maximum: (0, 0)  \n2  \n7) y = (2x − 8) 3  \nRelative minimum: (4, 0)  \nNo relative maxima.  \n4) y = x 4 − 2x2 + 3  \nRelative minima: (−1, 2), ( 1, 2)  \nRelative maximum: (0, 3)  \n6) y = −  2   \nx 2 − 4  \nRelative minimum: (0, ~~ ~~12)  \nNo relative maxima.  \n5 2  \n1  3   3  \n8) y = −  5  (x − 4) − 2(x − 4)  \nRelative minimum: (0, − ~~ ~~1252 )  \nRelative maximum: (4, 0)  \nCritical thinking questions:  \n9) Give an example function f (x) where f '' (0) = 0 and there is no relative minimum or maximum at x = 0.  \nMany answers. Ex: f (x) = 0, x, x 3 , etc  \n10) Give an example function f (x) where f '' (0) = 0 and there is a relative maximum at x = 0.  \nMany answers. Ex: f (x) = −x4  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaiaxaeNYZ0SvD","https://ap.wps.com/l/cbCaiaxaeNYZ0SvD","pdf",36260,"English","# Relative minima and maxima\n## Problems with provided solutions\n## Critical thinking questions (f''(0)=0)","[{\"question\":\"How are relative minima and maxima found on these worksheets?\",\"answer\":\"For each function, the worksheet asks for all points of relative minima and maxima and provides the corresponding coordinate pairs as answers.\"},{\"question\":\"What kinds of functions appear in the worksheet exercises?\",\"answer\":\"The problems focus on polynomial functions, including cubic and quartic forms, sometimes with shifts like (x − 4).\"},{\"question\":\"What do the critical thinking questions require?\",\"answer\":\"They ask for example functions f(x) such that f''(0)=0, with either no relative minimum/maximum at x=0 or a relative maximum at x=0, and the worksheet lists example answers such as f(x)=0 and f(x)=−x4.\"}]","Relative Extrema Worksheets - 04 - Relative Extrema | PDF",1791426260]